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__________________As long as notion is impossible because of partial usage of one's brain skills, new glasses will not help.
----
If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com )

__________________As long as notion is impossible because of partial usage of one's brain skills, new glasses will not help.
----
If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com )

__________________As long as notion is impossible because of partial usage of one's brain skills, new glasses will not help.
----
If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com )

By following after? The standard definitions are the only ones that matter. Any other meaning you might imagine for, say, limit wouldn't be limit any more. It would be something else. You cannot show Mathematics to be broken simply by changing the meaning of things.

Quote:

...what is your answer to this question?

The proof can be found in lots of places. Just about any decent mathematics text covering convergent sequences would consider geometric sequences of the form, AxR1-n. They would provide a more general result than you require.

__________________A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group.

__________________As long as notion is impossible because of partial usage of one's brain skills, new glasses will not help.
----
If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com )

"Text" would, in general, not be something that corresponds to a link. Be that as it may, you seem to like Wikipedia, so let's go with two similar links from there. How about Geometric progressionWP and Geometric seriesWP.

__________________A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group.

"Text" would, in general, not be something that corresponds to a link. Be that as it may, you seem to like Wikipedia, so let's go with two similar links from there. How about Geometric progressionWP and Geometric seriesWP.

I do not see any rigorous proof there, please guide to such proof.

__________________As long as notion is impossible because of partial usage of one's brain skills, new glasses will not help.
----
If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com )

"Text" would, in general, not be something that corresponds to a link. Be that as it may, you seem to like Wikipedia, so let's go with two similar links from there. How about Geometric progressionWP and Geometric seriesWP.

I do not see any rigorous proof there, please guide to such proof.

I'm sorry you are having trouble following these things. Focus on the "Infinite geometric series" section of the first article and the "Proof of convergence" of the second.

__________________A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group.

I'm sorry you are having trouble following these things. Focus on the "Infinite geometric series" section of the first article and the "Proof of convergence" of the second.

∞ does not rigorously define infinity, so since "Infinite geometric series" and "Proof of convergence" use it, they do not define any rigorous proof.

A rigorous proof must be based on infinity as defined by transfinite cardinality for example: aleph0 < aleph1 < aleph2 < ... etc.

__________________As long as notion is impossible because of partial usage of one's brain skills, new glasses will not help.
----
If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com )

∞ does not rigorously define infinity, so since "Infinite geometric series" and "Proof of convergence" use it, they do not define any rigorous proof.

Oh, I see. It is limits you don't understand. Odd that, since that is the very thing you have been trying to attack. Good luck with that. It is usually best to understand what you object to rather than simply object.

I take that back. It is not just limits you don't understand for the matter at hand. Infinite sequences escape your grasp, too. Otherwise, you'd know why your statement quoted here is, as they would say on the old Perry Mason TV series, incompetent, irrelevant, and immaterial.

__________________A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group.

It does not change the fact that a rigorous proof must be based on infinity as defined by transfinite cardinality for example: aleph0 < aleph1 < aleph2 < ... etc.

∞ does not provide the necessary rigorous terms exactly because it does not distinguish between aleph0, aleph1, aleph2, ... etc.

__________________As long as notion is impossible because of partial usage of one's brain skills, new glasses will not help.
----
If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com )

It does not change the fact that a rigorous proof must be based on infinity as defined by transfinite cardinality for example: aleph0 < aleph1 < aleph2 < ... etc.

∞ does not provide the necessary rigorous terms.

As I said, your failure to understand limits and sequences allows you to believe your objection has substance. It does not. Why it does not is basic to both concepts.

However, just so you don't get completely hung up on a pro forma objection, one without substance, perhaps you might consider the question of where would the proof break down if your objection did have substance. Put another way, what purpose does the infinity used in the proof serve, since at no point does infinity itself in any of its possible forms ever actually enter into the argument.

__________________A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group.

__________________As long as notion is impossible because of partial usage of one's brain skills, new glasses will not help.
----
If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com )

where would the proof break down if your objection did have substance.

There is no rigorous proof by using inaccurate form like ∞ for infinity.

Originally Posted by jsfisher

Put another way, what purpose does the infinity used in the proof serve, since at no point does infinity itself in any of its possible forms ever actually enter into the argument.

Infinity itself (notated by ∞ inaccurate form) actually enters into the argument, exactly as seen in "Infinite geometric series" section of Geometric progressionWP or in "Proof of convergence" section of Geometric seriesWP.

__________________As long as notion is impossible because of partial usage of one's brain skills, new glasses will not help.
----
If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com )

There is no rigorous proof by using inaccurate form like ∞ for infinity.

Good thing, then, the proof uses no form of infinity.

Quote:

Infinity itself (notated by ∞ inaccurate form) actually enters into the argument, exactly as seen in "Infinite geometric series" section of Geometric progressionWP or in "Proof of convergence" section of Geometric seriesWP.

The appearance of a particular symbol in a notation does not make "infinity" a part of the proof. Look at the definition for 'limit' as the domain variable tends to "infinity". No where does "infinity" of any form find its way into the definition, and no where does "infinity" of any form find its way into the proof.

If you think otherwise, go ahead and identify exactly where "infinity" enters into the prove (as something other than mere notation for 'limit').

__________________A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group.

Look at the definition for 'limit' as the domain variable tends to "infinity". No where does "infinity" of any form find its way into the definition, and no where does "infinity" of any form find its way into the proof.

Dear jsfisher are there finitely or infinitely many elements in sequence (0.910, 0.0910, 0.00910, ...) that all of them are used in order to construct 0.910+0.0910+0.00910+ ... = 0.999...10 (such that 0.999...10 is not less than the sum over all (0.910, 0.0910, 0.00910, ...) elements) ?

-------------------------

Once again:

for all ε > 0, there exists a natural number N in N such that for all n ≥ N, |an - L| < ε.

So |an - L| must be = 0 in order to conclude that the sum over an = L.

It must be stressed that if infinity is not involved here, then |an - L| > 0 and we can't conclude that the sum over an = L.

For example, where is the rigorous proof that the sum over all (0.910, 0.0910, 0.00910,...) = L = 1?

__________________As long as notion is impossible because of partial usage of one's brain skills, new glasses will not help.
----
If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com )

Dear jsfisher are there finitely or infinitely many elements in sequence (0.910, 0.0910, 0.00910, ...) that all of them are used in order to construct 0.910+0.0910+0.00910+ ... = 0.999...10 (such that 0.999...10 is not less than the sum over all (0.910, 0.0910, 0.00910, ...) elements) ?

Nowhere does that enter into the proof. Why to you harp on the irrelevant?

Quote:

...
for all ε > 0, there exists a natural number N in N such that for all n ≥ N, |an - L| < ε.

See? By your own words, no infinity anywhere to be found.

Quote:

So |an - L| must be = 0 in order to conclude that the sum over an = L.

No, this does not follow at all. The requirement is |an - L| < ε where ε > 0. It is perfectly acceptable for 0 < |an - L|.

Quote:

It must be stressed that if infinity is not involved here, then |an - L| > 0 and we can't conclude that the sum over an = L.

In fact, we can conclude that. It is a matter of definitions for those things you don't understand: sequences and limits.

__________________A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group.

Normally, the term infinite sequence refers to a sequence which is infinite in one direction, and finite in the other—the sequence has a first element, but no final element, it is called a singly infinite, or one-sided (infinite) sequence, when disambiguation is necessary.

Furthermore, since 0.999...10 is the sum over one-sided (infinite) sequence like (0.910, 0.0910, 0.00910, ...) where the number of its terms = |N|, 0.999...10 alone can't be equal to 1 and the solution is given in http://www.internationalskeptics.com...&postcount=316.

__________________As long as notion is impossible because of partial usage of one's brain skills, new glasses will not help.
----
If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com )

In that case it is perfectly acceptable that 0.999...10 < 1 by |an - L| > 0, or in other words

The two are unrelated. My observation is supported by the very definition of limit. Yours is not supported by anything.

Quote:

Quote:

for all ε > 0, there exists a natural number N in N such that for all n ≥ N, |an - L| < ε.

is not a rigorous proof that, for example, 0.999...10 = 1.

Of course not. The "for all..." is merely a part of the formal definition of limit. However, that definition comes into play in showing the limit of a particular sequence of partial sums is unity, and that establishes the value for a particular series, and that, in turn, establishes -- rigorously -- that 0.999... is identical to 1.

Normally, the term infinite sequence refers to a sequence which is infinite in one direction, and finite in the other—the sequence has a first element, but no final element, it is called a singly infinite, or one-sided (infinite) sequence, when disambiguation is necessary.

I did not ignore it at all.

Quote:

Furthermore, since 0.999...10 is the sum over one-sided (infinite) sequence like (0.910, 0.0910, 0.00910, ...) where the number of its terms = |N|, 0.999...10 alone can't be equal to 1 and the solution is given in http://www.internationalskeptics.com...&postcount=316.

And just how is "the sum over one-sided [sic] (infinite) sequence" determined? By definition, it is the limit of the sequence of partial sums.

Determining the limit of the sequence of partial sums does not involve infinity at any step.

__________________A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group.

Determining the limit of the sequence of partial sums does not involve infinity at any step.

And how many partial sums there are in (s1, s2, s3, ...) sequence, finitely many or infinitely many?

__________________As long as notion is impossible because of partial usage of one's brain skills, new glasses will not help.
----
If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com )

__________________As long as notion is impossible because of partial usage of one's brain skills, new glasses will not help.
----
If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com )

The very definition of limit, if it is limited to finite sequences, does not support the assertion that the sum over all terms of sequence (0.910, 0.0910, 0.00910) actually = 1.

Moreover, the sum of |N| terms of that sequence < 1 , and we need a gap closer at |R| size in order to actually define 1 as the limit of 0.999...10.

Originally Posted by jsfisher

The "for all..." is merely a part of the formal definition of limit. However, that definition comes into play in showing the limit of a particular sequence of partial sums is unity, and that establishes the value for a particular series, and that, in turn, establishes -- rigorously -- that 0.999... is identical to 1.

0.999...10 is not identical to 1, and the sequence of partial sums, even if it has |N| terms, does not establish -- rigorously -- that 0.999...10 is identical to 1, simply because without using also|R| size, 0.999...10 < 1 by exactly 0.000...110, where 0.000... is a place value keeper that the number of its places is in bijection with the number of places in 0.999... .

The number of places is at most at |N| size, so in order to actually close the gap between 0.999...10 and 1, ...110 that is at |R| size is needed to actually do the job.

In other words: 0.999...10 + 0.000...110 = 1.

More generally, the sum over an where an has at most |N| terms < L by sum at |R| size.

__________________As long as notion is impossible because of partial usage of one's brain skills, new glasses will not help.
----
If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com )

Let's clarify it by a given example that can be used without loss of generality:

The sum over (0.910, 0.0910, 0.00910, 0.00110) = 1

The sum over (0.910, 0.0910, 0.00910, 0.000910) < 1

Please pay attention that in this finite example the sum = 1 if we are using the smallest value 0.00110 that is at higher level than value 0.000910.

In the case of the sum over infinity many values like (0.910, 0.0910, 0.00910, 0.000910, ...) the smallest value > 0 (notated as 0.000...110) is determined as the sum at |R| size (notates here as ...110), which is at higher level than the sum at |N| size (notate here as 0.000..., and 0.000...110 is taken as a unique value exactly as 0.00110 is taken as a unique value.

__________________As long as notion is impossible because of partial usage of one's brain skills, new glasses will not help.
----
If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com )

According to the example above, one can understand that a sum over sequence with |R| terms (which can't be symbolized since it is uncountable) needs ...1 at |P(R)| size in order to be identical to 1, ... etc. ad infinitum, so number systems (whether they are countable or uncountable) are determined by different levels of resolutions, where the higher resolution is taken as one unique value (notated as ...1) from the point of view of the lower resolution, and it closes the gap between the limit value and the sum of the lower resolution.

On top of all this there is the non-composed highest level that is beyond all resolutions, and it closes the gap to 1 for all resolutions.

In other words, the non-composed highest level is independent of any point of view, in order to be used to close the gap between the sum of a given resolution, and 1.

So, that is beyond transfinite cardinality is indeed actual infinity, and it is invariant and objective to all levels of collections, that each one of them or all of them together are not invariant and objective as the non-composed level is.

----------------------------

If we use the point of view of Physics on this case, then Newtonian physics is useful even if it is not accurate as GRT, or in other words, by using this analogy, the relative different points of view of different resolutions are still useful in order to close the gap between some sum at smaller transfinite cardinality to some limit value, by using a given sum of bigger transfinite cardinality (which is equivalent here to Newtonian physics).

Yet the total solution is provided only by using the non-composed highest level that is beyond all resolutions (which is equivalent here to GRT, in this analogy).

----------------------------

Definitions like "for all ε > 0, there exists a natural number N in N such that for all n ≥ N, |an - L| < ε", simply block the ability to understand the relative view of different levels of cardinality (or the concept of relative levels, in general), and the absolute view of the non-composed level beyond all transfinite cardinalities, and how the relative or the absolute views are used in order to solve the limit problem.

__________________As long as notion is impossible because of partial usage of one's brain skills, new glasses will not help.
----
If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com )

Let's clarify it by a given example that can be used without loss of generality...

Let's not. You continue to show complete disregard for and lack of understanding of how things are defined. Despite your objections, the series for an infinite sequence is, by definition the limit of the sequence of partial sums. Determination of that limit does not at any point involve infinity, in any of its possible forms.

If you are unwilling to accept and comply with fundamental mathematical definition, then you are not doing Mathematics.

You, doronshadmi, are not doing Mathematics.

__________________A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group.

You, jsfisher, put arbitrary restrictions to mathematical development, and use them in a dogmatic way.

Originally Posted by jsfisher

If you are unwilling to accept and comply with fundamental mathematical definition, then you are not doing Mathematics.

If you are unwilling to re-examine fundamental mathematical definitions, then you are not doing Mathematics.

Originally Posted by jsfisher

Despite your objections, the series for an infinite sequence is, by definition the limit of the sequence of partial sums. Determination of that limit does not at any point involve infinity, in any of its possible forms.

__________________As long as notion is impossible because of partial usage of one's brain skills, new glasses will not help.
----
If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com )

You, jsfisher, put arbitrary restrictions to mathematical development, and use them in a dogmatic way.

As you have been invited many times in the past, if you want to explore new areas of Mathematics, do so. However, you are not allowed to redefine established terminology and declare prior results wrong as a consequence.

However, you are not allowed to redefine established terminology and declare prior results wrong as a consequence.

I can re-examine or re-define your so called "established terminology", and this is a perfectly valid way of mathematical development.

The difference between you and me is that since you are now a moderator, you can shut me up because I do not follow after your so called "established terminology".

But before you going to shut me up, be aware of the fact that this thread is a part of "Religion and Philosophy" forum, which somehow you became one of it moderators.

__________________As long as notion is impossible because of partial usage of one's brain skills, new glasses will not help.
----
If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com )

It is an irrelevant question and has no bearing on the valuation of a series.

It is relevant, but you choose to ignore this question because it does not fit to your so called "established terminology".

Again, if you are unwilling to re-examine fundamental mathematical definitions, then you are not doing Mathematics.

__________________As long as notion is impossible because of partial usage of one's brain skills, new glasses will not help.
----
If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com )

You are not examining them; to are totally disregarding them and substituting your own.

I do not follow after your arbitrary restriction of finitism on limits, which is no more than your philosophical approach about this subject, so?

__________________As long as notion is impossible because of partial usage of one's brain skills, new glasses will not help.
----
If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com )

where limit is not restricted only to finitism, and relative or absolute observations are used.

In all cases (depending on relative or absolute observations) a sum over a given sequence reaches the limit (and not just tends to, as defined by your finite-only observation).

__________________As long as notion is impossible because of partial usage of one's brain skills, new glasses will not help.
----
If a tree falls in the forest, and no one’s there to see it, the tree and ground still measure each other. ( http://www.askamathematician.com )

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